We explain some aspects of the relation between the geometry and topology
of hyperbolic 3-manifolds and the rank of their fundamental group, i.e.
the minimal number of elements needed to generate it. For instance we show
that for every $k$ there are only finitely many conmensurability classes
of non-compact arithmetic hyperbolic 3-manifolds whose fundamental group
has rank $k$. This is a joint work with Ian Biringer.